3.1753 \(\int \sqrt{a+\frac{b}{x}} x^{5/2} \, dx\)

Optimal. Leaf size=74 \[ \frac{16 b^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{8 b x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{35 a^2}+\frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{7 a} \]

[Out]

(16*b^2*(a + b/x)^(3/2)*x^(3/2))/(105*a^3) - (8*b*(a + b/x)^(3/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(3/2)*x^(7/
2))/(7*a)

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Rubi [A]  time = 0.021808, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{16 b^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{8 b x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{35 a^2}+\frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]*x^(5/2),x]

[Out]

(16*b^2*(a + b/x)^(3/2)*x^(3/2))/(105*a^3) - (8*b*(a + b/x)^(3/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(3/2)*x^(7/
2))/(7*a)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{a+\frac{b}{x}} x^{5/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{7/2}}{7 a}-\frac{(4 b) \int \sqrt{a+\frac{b}{x}} x^{3/2} \, dx}{7 a}\\ &=-\frac{8 b \left (a+\frac{b}{x}\right )^{3/2} x^{5/2}}{35 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{7/2}}{7 a}+\frac{\left (8 b^2\right ) \int \sqrt{a+\frac{b}{x}} \sqrt{x} \, dx}{35 a^2}\\ &=\frac{16 b^2 \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}{105 a^3}-\frac{8 b \left (a+\frac{b}{x}\right )^{3/2} x^{5/2}}{35 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{3/2} x^{7/2}}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.013109, size = 47, normalized size = 0.64 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b) \left (15 a^2 x^2-12 a b x+8 b^2\right )}{105 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]*x^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)*(8*b^2 - 12*a*b*x + 15*a^2*x^2))/(105*a^3)

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Maple [A]  time = 0.004, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 15\,{a}^{2}{x}^{2}-12\,xab+8\,{b}^{2} \right ) }{105\,{a}^{3}}\sqrt{x}\sqrt{{\frac{ax+b}{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(1/2)*x^(5/2),x)

[Out]

2/105*(a*x+b)*(15*a^2*x^2-12*a*b*x+8*b^2)*x^(1/2)*((a*x+b)/x)^(1/2)/a^3

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Maxima [A]  time = 0.977571, size = 70, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (15 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 42 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}} + 35 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2} x^{\frac{3}{2}}\right )}}{105 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)*x^(5/2),x, algorithm="maxima")

[Out]

2/105*(15*(a + b/x)^(7/2)*x^(7/2) - 42*(a + b/x)^(5/2)*b*x^(5/2) + 35*(a + b/x)^(3/2)*b^2*x^(3/2))/a^3

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Fricas [A]  time = 1.48451, size = 113, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (15 \, a^{3} x^{3} + 3 \, a^{2} b x^{2} - 4 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{105 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)*x^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*a^3*x^3 + 3*a^2*b*x^2 - 4*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x)/a^3

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Sympy [B]  time = 110.661, size = 314, normalized size = 4.24 \begin{align*} \frac{30 a^{5} b^{\frac{9}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac{66 a^{4} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac{34 a^{3} b^{\frac{13}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac{6 a^{2} b^{\frac{15}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac{24 a b^{\frac{17}{2}} x \sqrt{\frac{a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac{16 b^{\frac{19}{2}} \sqrt{\frac{a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(1/2)*x**(5/2),x)

[Out]

30*a**5*b**(9/2)*x**5*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 66*a**4*b**(11/
2)*x**4*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 34*a**3*b**(13/2)*x**3*sqrt(a
*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 6*a**2*b**(15/2)*x**2*sqrt(a*x/b + 1)/(105*
a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 24*a*b**(17/2)*x*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210
*a**4*b**5*x + 105*a**3*b**6) + 16*b**(19/2)*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*
b**6)

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Giac [A]  time = 1.1941, size = 68, normalized size = 0.92 \begin{align*} -\frac{2}{105} \,{\left (\frac{8 \, b^{\frac{7}{2}}}{a^{3}} - \frac{15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}}{a^{3}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)*x^(5/2),x, algorithm="giac")

[Out]

-2/105*(8*b^(7/2)/a^3 - (15*(a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 35*(a*x + b)^(3/2)*b^2)/a^3)*sgn(x)